The Platonic Solids: Why Only Five?
1
by James D. Nickel
There are technically an infinite number of regular convex plane figures (all angles equal and all sides
equal) or regular polygons. Below are the first six: equilateral (equal sides) triangle, square, regular pentagon
(five angles), regular hexagon (six angles), regular heptagon (seven angles), and regular octagon (eight angles).2
All corresponding parts of each of these regular convex figures are equal (i.e., all edges [or sides] and all
angles in each figure are equal). Note also that only two edges meet at each vertex of each figure.
Let’s now shift to the three-dimensional world. Since, in two-dimensions, there are an infinite number
of regular polygons, can we, using reasoning by analogy, conclude that, in three-dimensions, there are an
infinite number of regular polyhedra, a solid whose faces are all regular polygons? This conclusion about
three-dimensions “seems” to be a natural inference from the twodimensional situation. The way mathematicians think about three diNote: This essay is exmensional figures like regular polyhedra is to employ the methods of
tracted from a Lesson
one of its most interesting branches, topology.3 Using topology, we
from the forthcoming
can study this problem by investigating two unique properties of regutextbook Mathematics:
lar polyhedra: (1) the same number of edges bound each face and (2)
Building on Foundations.
the same number of edges meet at every vertex.
To illustrate, picture the cube (a regular polyhedron) at left. The cube has 8 vertices, 6 faces, and 12 edges where 4 edges bound each
face and 3 edges meet at each vertex.
Next, consider the tetrahedron (literally, “four
faces” where each face is an equilateral triangle). A
tetrahedron has 4 vertices, 4 faces, and 6 edges where
3 edges bound each face and 3 edges meet at each vertex.
Cube (Hexahedron)
Note that the two properties, (1) equal edges
bounding each face and (2) equal edges meeting at each vertex, have nothing to
do with size or shape. Here is where topological tools show us that only five regular
Tetrahedron
polyhedra can satisfy these two requirements.
We have noted two of these figures, the cube and the tetrahedron. The other three are the octahedron (8
faces; octa from the Greek means “eight”), dodecahedron (12 faces; dodeca from the Greek means “2 plus
10” or “twelve”), and icosahedron (20 faces; icosa from the Greek means “twenty”).
1 Convex is derived from the Latin convexus meaning “rounded.” In a convex polygon, the line segment connecting any two points
inside the polygon will always stay completely inside the polygon.
2 Note that the number of sides determines the measure of each angle. In general, given a regular polygon with n sides, A, the
measure of each angle, is determined by the following formula:
 n  2  180o
 4  2  180o 360o
. For a square, A 

 90o .
A
n
4
4
3 Topology is the study of those properties of geometric forms that remain invariant under certain transformations, as bending or
stretching.
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The Platonic Solids: Why Only Five?
by James D. Nickel
How many edges, faces, and vertices are there in a dodecahedron and an icosahedron, the two figures
on the left of the picture above? Instead of just counting (it is easy to get “lost” with these figures), we
might want to apply a more systematic approach. For example, we know that
the dodecahedron has 12 faces. How many edges on each face? 5. We could
calculate the product of 12 and 5 to get the number of edges, but there would
be a problem with this approach. How many faces does each edge share? 2.
12  5
 30 .
Therefore, we’ve got to divide the product of 12 and 5 by 2 and
2
Similarly, the face of each dodecahedron has five vertices. We could again calculate the product of 12 and 5 to get the number of vertices, but we would encounter the same problem as before; i.e., each vertex
shares a face. How many faces does each vertex share?
Icosahedron
3. To calculate the number of vertices in a dodecahe-
12  5
 20 . For the icosahedron,
3
there are 20 faces with each face having 3 edges and
each edge shares 2 faces. To calculate the number of
20  3
Octahedron
 30 .
edges in an icosahedron, we compute
2
Each of the 20 faces has 3 vertices and each vertex
shares 5 faces. To calculate the number of vertices in an icosahedron, we comDodecahedron
20  3
pute
 12 .
5
We tabulate our observations as follows (I would highly recommend that these solids be built; there are
many sources on the web that will provide templates or “nets” for building them):
dron, we compute
Table I
Regular polyhedra Shape of face
F (number of faces) V (number of vertices) E (number of edges)
tetrahedron
equilateral triangle
4
4
6
cube
square
6
8
12
octahedron
equilateral triangle
8
6
12
dodecahedron regular pentagon
12
20
30
icosahedron
equilateral triangle
20
12
30
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The Platonic Solids: Why Only Five?
by James D. Nickel
There is a discernible pattern relating the number of faces, vertices, and edges in each regular polyhedron. Can you figure it out? The Swiss mathematician Leonhard Euler (1707-1783), the founder of topology, did. Here is the formula (called the Euler characteristic):
F + V = E + 2 or V – E + F = 2
Topology tells us that we cannot reason as we did from an infinite number of regular polygons in twodimensions to an infinite number of regular polyhedra in three-dimensions. We noted that in twodimensions, only two edges meet at each vertex of each figure. In three-dimensions, you can have any number
of edges meet at each vertex and, furthermore, any number of faces. For example, you could have 50 edges
meet at a vertex and 6 at another. One face could be a square while another face could be a 40-sided polygon. To restrict a solid figure to equal edges bounding each face and equal edges meeting at each vertex confines the number of such figures to five.
There are several ways to prove that there are only five Platonic solids.4 Note, that mathematical proof
does not prove anything by exercising rational, independent thought (although this “independence” is assumed by unbelieving mathematicians). Mathematical proof is one way in which man uses logical reasoning,
a gift from the Creator God, to justify both visible and invisible patterns of God’s law-ordered creation.
That there are only five Platonic solids is one of those given patterns.
The proof will employ Euler’s characteristic:
Equation 1: V – E + F = 2
Let’s consider a given polyhedron where s iw the number of sides in each of the faces. It will have F
identical faces. Based upon our table, the value of F can be 4, 6, 8, 12, or 20. Counting the total number of
edges, we get:
Equation 2. sF = 2E
Why? Since each edge belongs to two faces, it is therefore connected twice to sF. We then solve for F.
We get:
Equation 3. F =
2E
s
Let’s let r = the number of edges that meet at each vertex V. For example, for the octahedron, V = 6
and r = 4. Counting the total number of edges again, we get:
The Platonic Solids are named after Plato (427-347 BC), the ancient Greek philosopher who was intrigued by the threedimensional symmetries revealed in these shapes.
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The Platonic Solids: Why Only Five?
by James D. Nickel
Equation 4. rV = 2E
Again, since each edge connects two vertices, it is therefore connected twice to rV. We then solve for
V. We get:
Equation 5. V =
2E
r
Substituting Equations 3 and 5 into Equation 1, we get:
2E
2E
E
2
Equation 6.
r
s
Using algebra, we can translate Equation 6 into Equation 9 as follows:
2
2 2
1 
(divide both members of Equation 6 by E)
r
s E
1 1 1 1
Equation 8.   
(divide both members of Equation 7 by 2)
r 2 s E
1
1 1 1 1
Equation 9.    (add to members of Equation 8)
r s E 2
2
Equation 7.
Since a polygon must have at least three sides and since at least three edges must meet at each vertex of
a polyhedron, then s  3 and r  3.
Study Equation 9. What is it revealing about r and s? If s and r are both greater than 3, we have a contra1 1 1 1
1 1 1
1
diction. For example, let s = 4 and r = 4. Then, we have        0 
E=
4 4 E 2
2 E 2
E
0, a contradiction based upon the given parameters of the figures (to make sense, E must be greater than 0).
1 1 1 1
9
1 1
1
1
If s = 5 and r = 4, then    
 E = -20, another contradiction
    
5 4 E 2
20 E 2
20 E
based upon the given parameters of the figures (E > 0). By this reasoning, we only have to find the possible
values of r when s = 3 and s when r = 3.
Setting s = 3 in Equation 9, we get:
1 1 1 1
1 1 1
1 1 1
      
 
r 3 E 2
r E 6
E r 6
1 1 1
  , is true only if r = 3, 4, or 5. If r = 6, then E = 0, a contradiction. If r > 6, the
E r 6
E < 0, a contradiction. Note the corresponding values of E:
This equation,
Table II
r (edges) E Regular polyhedra
3
6 tetrahedron
4
12 octahedron
5
30 icosahedron
Note that the values of E in Table II account for three regular polyhedra in Table I.
Now, let’s set r = 3 in Equation 9. We get:
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The Platonic Solids: Why Only Five?
by James D. Nickel
1 1 1 1
1 1 1
1 1 1
      
 
s E 6
E s 6
3 s E 2
We get the same result in terms of s. This is so because Equation 9 is symmetric with regards to the variables s and r meaning these variables can be swapped. Hence, we generate a similar table:
Table III
s (sides) E Regular polyhedra
3
6 tetrahedron
4
12 cube
5
30 dodecahedron
Note also, because of this symmetry, these solids are the duals of
the ones obtained when s = 3. Duality is a geometrical principle that
shows the symmetry between two figures when their parts are interchanged. By the duality principle, for every polyhedron, there exists
another polyhedron in which faces and polyhedron vertices occupy
complementary locations. This polyhedron is known as the dual, or
reciprocal. The tetrahedron is the dual of itself, the octahedron is the
dual of the cube (and vice versa), and the icosahedron is the dual of the
dodecahedron (and vice versa).
Table IV
r or s E Regular polyhedra (r)
3
6 tetrahedron
4
12 octahedron
5
30 icosahedron
Tetrahedron, dual of itself
Dual (s)
tetrahedron
cube
dodecahedron
These cases exhaust all of the possibilities. Thus, although there are infinitely many regular polygons in a two-dimensional plane, there are only
five regular solids in three-dimensional space. QED.
What is even more amazing
about the reality that there are only
five regular solids in space is that
many small details of God’s creation
reflect this restriction; i.e., it is a creational covenant. Most crystals grow
in the beautiful shapes of regular
polyhedra. For example, sodium
Octahedron, dual of the Cube
chlorate crystals appear in the shape
of cubes and tetrahedra, while chrome alum crystals are in the form of
octahedra. Equally fascinating are the appearance of decahedra and
idosahedra crystals in the skeletal structures of radiolaria (microscopic
Amethyst Crystal (iStockPhoto)
sea animals).
Minerals, created by God as a gift to us, are technically defined as homogeneous5 inorganic6 solid substances having a definite chemical composition and characteristic crystalline structure, color, and hardness.
5
6
Homogeneous, from the Greek, means “same kind or same nature.”
Inorganic means “non-living” as opposed to organic which means “living.”
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The Platonic Solids: Why Only Five?
by James D. Nickel
Mineralogists study crystals, not only for their intrinsic beauty, but because these structures provide clues to
the arrangement of atoms within a mineral thus offering an important means of identification. Only a few
minerals, such as opal and silica glass, lack a crystal structure.
Scientists have discovered that all crystals can be placed in one of seven categories called systems. These
seven systems are classified as (1) Cubic system, (2) Hexagonal system, (3) Rhombohedral7 system, (4) Tetragonal8 system, (5) Orthorhombic9 system, (6) Monoclinc10 system, and (7) Triclinic11 system. These seven
systems can be further subdivided into thirty-two symmetry12 classes and 230 space-groups based on internal
structure.13 Treatment of these subdivisions constitutes the
study of crystallography.
Note the observations made by mathematician James
R. Newman (1907-1966):
Halite cubical system
In the development of crystallography the mathematics
of group theory and of symmetry has ... played a remarkable part ... mathematicians by an exhaustive logical analysis of certain properties of space and of the
Beryl hexagonal system
possible transformations (motions) within space [i.e.,
topology – JN], decreed the permissible variations of internal
structure of crystals before observers were able to discover their actual structure [italics added – JN]. Mathematics, in other
words, not only enunciated the applicable physical laws,
but provided an invaluable syllabus or research to guide
Idocrase tetragonal system
future experimenters. The history of the physical sciences contains many similar instances of mathematical prevision.14
Gypsum monoclinic
system
Quartz monoclinic system
Chlorite in quartz
Carefully reread these observations. Let’s unpack them from a Biblical viewpoint. Note first that scientists (crystallographers) discovered what mathematicians, by logical analysis, a preveniently decreed. Before
these scientists began their explorations, mathematicians had already proved that the symmetry elements of
crystals could only be grouped in 32 different ways, and no others. Here we see that mathematicians, by logical
analysis alone, blazed a trail for crystallographers to follow. In their research of crystals, these scientists confirmed the conclusions of the mathematicians. This fact reflects upon the remarkable connectivity between
man’s mind (his ability to reason) and the physical world. The mind of man, with his mathematical capabilities, and the physical world, with its mathematical properties, cohere because of a common Creator. Hence, we
should not be surprised to find a multitude of instances where mathematical conclusions not only connect
with, but also direct the physical sciences (just as Newman summarizes in the last two sentences of the above
quote).
A rhombohedron is a solid bounded by six rhombic planes where a rhombus is an oblique-angled equilateral parallelogram; i.e.,
a rhombus is any equilateral parallelogram except a square.
8 A tetragon is a polygon having four angles or four sides.
9 Ortho means “straight” or “right angled.”
10 Monoclinic literally means “one lean.”
11 Triclinic literally means “three leans.”
12 We have already used symmetry two times in this article (algebraic symmetry and topological duality). Symmetry, from the
Greek, means “like measure.” For a figure to reflect symmetry means that you can determine an exact correspondence of form
and constituent configuration on opposite sides of a dividing line or plane or about a center or an axis. Note that butterfly wings
are symmetric (the axis of symmetry is the body of the butterfly).
13 See Philippe Le Corbeiller, “Crystals and the Future of Physics,” in Scientific American (January 1953), 50-55.
14 James R. Newman, The World of Mathematics (New York: Simon and Schuster, 1956), 2:852-853.
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